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Pólya sequences, Toeplitz kernels and gap theorems. (English) Zbl 1204.30018

A sequence \(\Lambda\) of real numbers is called a Pólya sequence if an entire function of zero exponential type bounded on \(\Lambda\) must be constant. The authors solve the Pólya-Levinson problem to describe the Pólya sequences. It is also shown that the Pólya-Levinson problem is equivalent to the Beurling gap problem on Fourier transforms of measures. The main tools are Toeplitz kernels and de Branges spaces of entire functions.

MSC:

30D15 Special classes of entire functions of one complex variable and growth estimates
47B35 Toeplitz operators, Hankel operators, Wiener-Hopf operators
42A38 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
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